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Another vote for the Nikon 70-300mm VR (full frame version). Stays sharp out to 220mm, at 300mm it's still good. AF speed and VR are also good in my opinion. In the year 2006+13 you can find them cheap s/h. In good light it's great. For Astro it won't cut it though, that's a whole different kettle of fish.
Pic related is a bad example, missed critical focus on the eye. Shot with an older 16 mp camera.(d7000). well within your budget with the lens.
